How to apply the replica method to the Sherrington-Kirkpatrick model¶

The Sherrington-Kirkpatrick model is defined as follows:

$$ H = - \sum_{i < j} J_{ij} S_iS_j - h \sum_{i}S_i $$

$$ P \left( J _ { i j } \right) = \frac { 1 } { J } \sqrt { \frac { N } { 2 \pi } } \exp \left\{ - \frac { N } { 2 J ^ { 2 } } \left( J _ { i j } - \frac { J _ { 0 } } { N } \right) ^ { 2 } \right\} $$

$$ \left[ J _ { i j } \right] = \frac { J _ { 0 } } { N } , \quad \left[ \left( \Delta J _ { i j } \right) ^ { 2 } \right] = \frac { J ^ { 2 } } { N } $$

The partition function of this system is:

$$ \begin{align} Z &=& \mathrm{Tr}_\mathbf{s_{1:N}} \exp \left( \beta H\right) \\ &=& \underline{\mathrm{Tr}_\mathbf{s_{1:N}}} \exp \left( \beta \sum_{i<j} J_{ij} s_is_j + \beta h \sum_i s_i\right) \\ &\equiv& \underline{\sum_{s_1=\pm 1}\sum_{s_2=\pm 1}\dots \sum_{s_N=\pm 1}} \exp \left( \beta \sum_{i<j} J_{ij} s_is_j + \beta h \sum_i s_i\right) \end{align} $$

One could calculate free energy $F$:

$$ F = -T \log Z $$

Configurational average (i.e., average over all possible configuration of $\mathbf{J}$) of the $F$:

$$ \begin{align} [F] \equiv \int \prod_{(i, j)} dJ_{ij} P(J_{ij}) F = -T \int \prod_{(i, j)} dJ_{ij} P(J_{ij}) \log Z \\ = -T \int \prod_{(i, j)} dJ_{ij} P(J_{ij}) \log \left(\sum_{s_1=\pm 1}\sum_{s_2=\pm 1}\dots \sum_{s_N=\pm 1} \exp \left( \beta \sum_{i<j} J_{ij} s_is_j + \beta h \sum_i s_i\right)\right) \end{align} $$

Calculation of the $[F]$ in quite tough as the dependence of $\log Z$ on $\mathbf{J}$ is very complicated. Hence, we would like to proxy the evaluation by:

$$ [\log Z] = \lim_{n \rightarrow 0} \frac{[Z^n] - 1}{n} $$

which is called replica method(replica trick).

$$ \begin{align} Z^n &=& \left(\mathrm{Tr}_\mathbf{s_{1:N}} \exp \left( \beta H\right)\right)^n \\ &=& \left(\mathrm{Tr}_\mathbf{s_{1:N}} \exp \left( \beta \sum_{i<j} J_{ij} s_is_j + \beta h \sum_i s_i\right)\right)^n \\ &=& \left(\mathrm{Tr}_\mathbf{s_{1:N}^1} \exp \left( \beta \sum_{i<j} J_{ij} s_is_j + \beta h \sum_i s_i\right)\right) \times \\ & & \left(\mathrm{Tr}_\mathbf{s_{1:N}^2} \exp \left( \beta \sum_{i<j} J_{ij} s_is_j + \beta h \sum_i s_i\right)\right) \times \\ &\dots& \\ & & \times \left(\mathrm{Tr}_\mathbf{s_{1:N}^n} \exp \left( \beta \sum_{i<j} J_{ij} s_is_j + \beta h \sum_{i=1}^N s_i\right)\right)\\ &=& \left(\mathrm{Tr}_\mathbf{s_{1:N}^{1:n}} \exp \left( \beta \sum_{i<j}\sum_{\alpha=1}^n J_{ij} s_i^\alpha s^\alpha_j + \beta h \sum_{i=1}^N \sum_{\alpha=1}^n s^\alpha_i\right)\right) \end{align} $$

$\therefore$ $$ \begin{align} [Z^n] &=& \int \prod_{(i, j)} dJ_{ij} P(J_{ij})\left(\underline{\mathrm{Tr}_\mathbf{s_{1:N}^{1:n}}} \exp \left( \beta \sum_{i<j}\sum_{\alpha=1}^n J_{ij} s_i^\alpha s^\alpha_j + \underline{\beta h \sum_{i=1}^N \sum_{\alpha=1}^n s^\alpha_i}\right)\right) \\ &=& \underline{\mathrm{Tr}_\mathbf{s_{1:N}^{1:n}}} \underline{\exp{\left( \beta h \sum_{i=1}^N \sum_{\alpha=1}^n s^\alpha_i \right)}} \color{blue}{\int \prod_{i<j} dJ_{ij} P(J_{ij})\left( \exp \left( \beta \sum_{\alpha=1}^n J_{ij} s_i^\alpha s^\alpha_j\right)\right)} \end{align} $$

$$ \begin{align} &\color{blue}{\int \prod_{i<j} dJ_{ij} P(J_{ij})\left( \exp \left( \beta \sum_{i<j}\sum_{\alpha=1}^n J_{ij} s_i^\alpha s^\alpha_j\right)\right)} \\ &= \color{blue}{\int \prod_{i<j} dJ_{ij} \frac{1}{J}\sqrt{\frac{N}{2\pi}}} \color{green}{\left(\exp \left( -\frac{N}{2J^2}\left( J_{ij}-\frac{J_0}{N} \right)^2 + \beta \sum_{i<j}\sum_{\alpha=1}^n J_{ij} s_i^\alpha s^\alpha_j\right)\right)} \\ \end{align} $$

$$ \begin{align} & &\color{green}{\exp \left( -\frac{N}{2J^2}\left( J_{ij}-\frac{J0}{N} \right)^2 + \beta \sum{i<j}\sum{\alpha=1}^n J{ij} s_i^\alpha s^\alphaj\right)} \ &=& \color{green}{ \exp \left[ -\frac{N}{2J^2}\left( J{ij}^2 - \frac{2}{N}\left(J0 + J^2\beta \sum{\alpha=1}^n J_{ij} s_i^\alpha s^\alphaj\right)J{ij} + \left(\frac{J0}{N}\right)^2 \right)\right] } \ &=& \color{green}{ \exp \left[ -\frac{N}{2J^2} \left{ \left[ J{ij} - \frac{1}{N}\left(J0 + J^2\beta \sum{\alpha=0}^n s_i^\alpha s^\alpha_j \right) \right]^2

        - \frac{2}{N^2} J_0 J^2 \beta \sum_{\alpha=0}^n s_i^\alpha s^\alpha_j -\frac{J^4\beta^2}{N^2}
        \underline{\left(\sum_{\alpha=1}^n s_i^\alpha s^\alpha_j\right)^2}
\right\}

\right] } \ &=& \color{green}{ \exp \left[ -\frac{N}{2J^2} \left{ \left[ J_{ij} - \frac{1}{N}\left(J0 + J^2\beta \sum{\alpha=0}^n s_i^\alpha s^\alpha_j \right) \right]^2

        - \frac{2}{N^2} J_0 J^2 \beta \sum_{\alpha=0}^n s_i^\alpha s^\alpha_j -\frac{J^4\beta^2}{N^2}
        \underline{\sum_{\alpha, \gamma} s_i^\alpha s^\alpha_j s_i^\gamma s^\gamma_j}
\right\}

\right] } \end{align} $$

$\therefore$ $$ \begin{align} [Z^n] &=& \mathrm{Tr}\mathbf{s{1:N}^{1:n}} \exp{\left( \beta h \sum{i=1}^N \sum{\alpha=1}^n s^\alphai \right)} \color{blue}{\int \prod{i<j} dJ{ij} \frac{1}{J}\sqrt{\frac{N}{2\pi}}} \color{green}{ \exp \left[ -\frac{N}{2J^2} \left{ \left[ J{ij} - \frac{1}{N}\left(J0 + J^2\beta \sum{\alpha=0}^n s_i^\alpha s^\alpha_j \right) \right]^2

        - \frac{2}{N^2} J_0 J^2 \beta \sum_{\alpha=0}^n s_i^\alpha s^\alpha_j -\frac{J^4\beta^2}{N^2}
        \sum_{\alpha, \gamma} s_i^\alpha s^\alpha_j s_i^\gamma s^\gamma_j
\right\}

\right] } \ &=& \mathrm{Tr}\mathbf{s{1:N}^{1:n}} \exp{\left( \beta h \sum{i=1}^N \sum{\alpha=1}^n s^\alphai \right)} \int \prod{i<j} dJ{ij} \frac{1}{J}\sqrt{\frac{N}{2\pi}} \exp \left[ -\frac{N}{2J^2} \left{ \left[ J{ij} - \frac{1}{N}\left(J0 + J^2\beta \sum{\alpha=0}^n s_i^\alpha s^\alpha_j \right) \right]^2 \underline{- \frac{2}{N^2} J0 J^2 \beta \sum{\alpha=0}^n s_i^\alpha s^\alphaj -\frac{J^4\beta^2}{N^2} \sum{\alpha, \gamma} s_i^\alpha s^\alpha_j s_i^\gamma s^\gammaj} \right} \right] \ &=& \mathrm{Tr}\mathbf{s{1:N}^{1:n}} \exp{\left( \beta h \sum{i=1}^N \sum_{\alpha=1}^n s^\alphai \right)} \prod{i<j} \underline{\exp\left[

- \frac{J_0\beta}{N} \sum_{\alpha=0}^n s_i^\alpha s^\alpha_j
-\frac{J^2\beta^2}{2N}\sum_{\alpha, \gamma} s_i^\alpha s^\alpha_j s_i^\gamma s^\gamma_j

\right]} \ \underbrace{\int dJ{ij} \frac{1}{J}\sqrt{\frac{N}{2\pi}} \exp \left[ -\frac{N}{2J^2} \left{ \left[ J{ij} - \frac{1}{N}\left(J0 + J^2\beta \sum{\alpha=0}^n s_i^\alpha s^\alphaj \right) \right]^2 \right} \right] }{=1} \ &=& \mathrm{Tr}\mathbf{s{1:N}^{1:n}} \underline{\exp{\left( \beta h \sum{i=1}^N \sum{\alpha=1}^n s^\alphai \right)}} \prod{i<j} \underline{\exp\left[

- \frac{J_0\beta}{N} \sum_{\alpha=0}^n s_i^\alpha s^\alpha_j
-\frac{J^2\beta^2}{2N}\sum_{\alpha, \gamma} s_i^\alpha s^\alpha_j s_i^\gamma s^\gamma_j

\right]} \ &=& \mathrm{Tr}\mathbf{s{1:N}^{1:n}} \prod_{i<j} \underline{\exp\left[ \color{red}{\frac{1}{N}\left(

     - \frac{J^2\beta^2}{2}\sum_{\alpha, \gamma} s_i^\alpha s^\alpha_j s_i^\gamma s^\gamma_j
     - \beta J_0 \sum_{\alpha=0}^n s_i^\alpha s^\alpha_j
\right)}
+ \beta h \sum_{i=1}^N \sum_{\alpha=1}^n s^\alpha_i

\right]} \end{align} $$

$$ \begin{align} \exp\left[

- \frac{J_0\beta}{N^2} \beta \sum_{\alpha=0}^n s_i^\alpha s^\alpha_j
-\frac{J^2\beta^2}{2N}\sum_{\alpha, \gamma} s_i^\alpha s^\alpha_j s_i^\gamma s^\gamma_j

\right] \int \prod{i<j} dJ{ij} \frac{1}{J}\sqrt{\frac{N}{2\pi}} \exp \left[ -\frac{N}{2J^2} \left{ \left[ J_{ij} - \frac{1}{N}\left(J0 + J^2\beta \sum{\alpha=0}^n s_i^\alpha s^\alpha_j \right) \right]^2 \right} \right] \end{align} $$

$\color{blue}{x} y$

$$ \left[ Z ^ { n } \right] = \int \left( \prod _ { i < j } \mathrm { d } J _ { i j } P \left( J _ { i j } \right) \right) \operatorname { Tr } \exp \left( \beta \sum _ { i < j } J _ { i j } \sum _ { \alpha = 1 } ^ { n } S _ { i } ^ { \alpha } S _ { j } ^ { \alpha } + \beta h \sum _ { i = 1 } ^ { N } \sum _ { \alpha = 1 } ^ { n } S _ { i } ^ { \alpha } \right) $$

$$ \operatorname { Trexp } \left\{ \frac { 1 } { N } \sum _ { i < j } \left( \frac { 1 } { 2 } \beta ^ { 2 } J ^ { 2 } \sum _ { \alpha , \beta } S _ { i } ^ { \alpha } S _ { j } ^ { \alpha } S _ { i } ^ { \beta } S _ { j } ^ { \beta } + \beta J _ { 0 } \sum _ { \alpha } S _ { i } ^ { \alpha } S _ { j } ^ { \alpha } \right) + \beta h \sum _ { i } \sum _ { \alpha } S _ { i } ^ { \alpha } \right\} $$